The Octagonal PET II: The Topology of the Limit Sets

TL;DR

This paper classifies the topological structure of limit sets in a specific dynamical system using renormalization, identifying conditions under which the limit set is a finite forest or a Cantor set, with explicit parameter criteria.

Contribution

It extends previous work by applying renormalization to classify limit sets' topology in the system, providing explicit parameter conditions for different topological types.

Findings

Limit set is either a finite forest or a Cantor set.

Explicit parameter conditions determine the topology of the limit set.

Special case: limit set is two disjoint arcs if continued fraction coefficients are even at odd positions.

Abstract

This is a sequel to my paper "The Octagonal PET I: Renormalization and Hyperbolic Symmetry". In this paper we use the renormalization scheme found in the first paper to classify the limit sets of the systems according to their topology. The main result is that the limit set is either a finite forest or a Cantor set, with an explicit description of which cases occur for which parameters. In one special case, the limit set is a disjoint union of 2 arcs if and only if the continued fraction expansion of the parameter has the form [a0:a1:a2:a3...] with a_k even for every odd k.